$10.$ Consider the low$-$pass filter in the following circuit, which has a load resistor RL$.$ L $+$ v$\_\{$o$\}+$ v$\_\{$i$\}$ R$\_\{$L$\}$ H$($s$)=($V$\_\{$o$\}($s$))/($V$\_\{$i$\}($s$))$ a$)$ What is the transfer function of the unloaded filter? b$)$ What is the transfer function of the loaded filter? c$)$ Compare the transfer function of the unloaded filter in part $($a$)$ and the transfer function of the loaded filter in part $($b$).$ Are the cutoff frequencies different? Are the passband gains different? d$)$ If R $=1$ ko and L $=20$ mH$,$ what is the cutoff frequency of the unloaded filter in rad$/$s$?$ Answers: a$)$ HUL$($JW$)$: R$/$L H L $($j omega$)=$ R L $($ R L R$+$R L $)$ j$*$omega $+$ R$/$L $*($R$/($R $+$ R$\_\{$L$\}))^$ i $)$ omega c UL $=$ R L $,$ omega CL $=$ R L $($ R L R$+$R L $)$; d$)50,000$ rad$/$s